Finite Group Databasebeta

← #11722869

Subgroups — #11722869

Order 864.

364 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 364 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#89021C211yesmaximal, characteristic, complemented
2C3×D3×D12C211yesmaximal, characteristic, complemented
3#88810C211yesmaximal, characteristic, complemented
4#88796C211yesmaximal, characteristic, complemented
5#88794C211yesmaximal, characteristic, complemented
6C3×#3714C211yesmaximal, characteristic, complemented
7C3×#3711C211yesmaximal, characteristic, complemented
8#63932—31nomaximal
9#63830—31nomaximal
10#63795—31nomaximal
11C2×C3×D32—21no—
12#6308C2211yescharacteristic, complemented
13C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C3×C4⋊(C3⋊D3)C2211yescharacteristic, complemented
15C32×D12C2211yescharacteristic
16C32×C4×D3C2211yescharacteristic
17#6290—21no—
18Dic3×C3⋊D3—21no—
19C3×#626—21no—
20C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
21C32×C3⋊C8C2211yescharacteristic, complemented
22#3828—31no—
23C3×D3×D4—31no—
24C2×C3×D12—31no—
25D3×D12D311yescharacteristic, complemented
26#3795—31no—
27#3795—31no—
28#3795—61no—
29#3754—31no—
30#3744—31no—
31C3×#213—31no—
32C3×#208—31no—
33C3×D24—31no—
34C3×ΓL(1,𝔽25)—31no—
35#3716—31no—
36#3716—31no—
37#3716—31no—
38#3716—31no—
39#3716—61no—
40#3716—61no—
41#3714—31no—
42#3714—31no—
43#3714D311yescharacteristic, complemented
44#3713—31no—
45#3711D311yescharacteristic, complemented
46C2×C3×C3⋊D3—21no—
47C2×C3×C3⋊D3—21no—
48C2×C32×D3—21no—
49C2×C32×D3D411yescharacteristic
50C3×D32—42no—
51C33×C4C2311yescharacteristic
52C3⋊(C3⋊Dic3)—21no—
53C32×Dic3D411yescharacteristic
54#932—91no—
55#911—91no—
56#908—91no—
57C22×C3×D3—61no—
58C22×C3×D3—61no—
59C2×D32—21no—
60C32×D4—31no—
61C2×C32×C4—31no—
62#638—61no—
63C4⋊(C3⋊D3)C2×D311yescharacteristic
64C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
65C4×C3⋊D3—61no—
66#634—31no—
67#634—31no—
68#634—31no—
69#634—61no—
70#634—61no—
71#634—61no—
72#634—121no—
73C3×#67—61no—
74C3×D12—21no—
75C3×D12—31no—
76C3×D12—31no—
77C3×D12—31no—
78C3×D12—31no—
79C3×D12—61no—
80C3×D12—61no—
81C3×D12—61no—
82C3×D12C2×D311yescharacteristic
83C3×D12C2×D311yescharacteristic
84C3×C4×D3—21no—
85C3×C4×D3C2×D311yescharacteristic
86C3×C4×D3C2×D311yescharacteristic
87#626—21no—
88#625—62no—
89#625—61no—
90#625—61no—
91D3×Dic3—62no—
92D3×Dic3—61no—
93D3×Dic3—61no—
94C32×C8—31no—
95C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
96C3×C3⋊C8—21no—
97C3×C3⋊C8—31no—
98C3×C3⋊C8—31no—
99C3×C3⋊C8—61no—
100C3×C3⋊C8C2×D311yescharacteristic, complemented
101C3×C3⋊C8C2×D311yescharacteristic, complemented
102C2×C33C2×D411yescharacteristic
103C3×C3⋊D3—41no—
104C3×C3⋊D3—41no—
105C32×D3—21no—
106C32×D3—41no—
107C2×C3×D4—91no—
108#237—91no—
109D3×D4—31no—
110#235—91no—
111#235—91no—
112#235—181no—
113C2×D12—31no—
114C3×D8—91no—
115C3×M16—91no—
116#214—91no—
117#214—91no—
118#214—91no—
119#214—181no—
120#213—31no—
121#213—91no—
122#213—91no—
123#208—31no—
124D24—31no—
125D24—91no—
126#204—91no—
127#204—91no—
128#204—91no—
129#204—181no—
130ΓL(1,𝔽25)—31no—
131C22×C32—31no—
132C22×C32—61no—
133C2×C3⋊D3—21no—
134C2×C3⋊D3—21no—
135C2×C3×D3—21no—
136C2×C3×D3—21no—
137C2×C3×D3—21no—
138C2×C3×D3—41no—
139C2×C3×D3—62no—
140C2×C3×D3—61no—
141C2×C3×D3—61no—
142C2×C3×D3—61no—
143C2×C3×D3—61no—
144C2×C3×D3—121no—
145C2×C3×D3—122no—
146C2×C3×D3—122no—
147C2×C3×D3D1211yescharacteristic
148C2×C3×D3#6711yescharacteristic
149D32—42no—
150C32×C4—21no—
151C32×C4—21no—
152C32×C4—21no—
153C32×C4—31no—
154C32×C4—41no—
155C32×C4C22×D311yescharacteristic
156C32×C4C22×D311yescharacteristic
157C32×C4C22×D311yescharacteristic
158C3⋊Dic3—61no—
159C3⋊Dic3—62no—
160C3⋊Dic3—61no—
161C3⋊Dic3—61no—
162C3⋊Dic3—121no—
163C3⋊Dic3—121no—
164C3⋊Dic3—122no—
165C3×Dic3—21no—
166C3×Dic3D1211yescharacteristic
167C3×Dic3#6711yescharacteristic
168AGL(1,ℤ/8)—271noSylow
169C33AGL(1,ℤ/8)11yescharacteristic, complemented, Sylow
170C23×C3—181no—
171C22×D3—61no—
172C22×D3—61no—
173C3×D4—31no—
174C3×D4—31no—
175C3×D4—61no—
176C3×D4—91no—
177C3×D4—91no—
178C3×D4—181no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—61no—
182#67—61no—
183#67—181no—
184#67—181no—
185#67—182no—
186#67—181no—
187C2×Dic3—181no—
188D12—31no—
189D12—31no—
190D12—31no—
191D12—31no—
192D12—61no—
193D12—61no—
194D12—61no—
195D12D3211yescharacteristic
196C4×D3—181no—
197C4×D3—181no—
198C4×D3—182no—
199C4×D3D3211yescharacteristic
200Dic6—91no—
201Dic6—91no—
202Dic6—91no—
203Dic6—181no—
204Dic6—181no—
205Dic6—181no—
206Dic6—361no—
207C3×C8—31no—
208C3×C8—31no—
209C3×C8—61no—
210C3×C8—91no—
211C3⋊C8—31no—
212C3⋊C8—31no—
213C3⋊C8—61no—
214C3⋊C8D3211yescharacteristic, complemented
215C2×C32—21no—
216C2×C32—21no—
217C2×C32—21no—
218C2×C32—41no—
219C2×C32—61no—
220C2×C32—121no—
221C2×C32C2×D1211yescharacteristic
222C2×C32D3×D411yescharacteristic
223C2×C32C2×#6711yescharacteristic
224C3⋊D3—41no—
225C3⋊D3—41no—
226C3×D3—21no—
227C3×D3—21no—
228C3×D3—41no—
229C3×D3—41no—
230C3×D3—41no—
231C3×D3—81no—
232C3×D3—122no—
233C3×D3—121no—
234C3×D3—121no—
235C3×D3—121no—
236C3×D3—121no—
237C3×D3—241no—
238#41—271no—
239C2×D4—91no—
240SD16—271no—
241SD16—271no—
242D8—91no—
243D8—271no—
244M16—91no—
245C22×C3—31no—
246C22×C3—31no—
247C22×C3—61no—
248C22×C3—61no—
249C22×C3—61no—
250C22×C3—121no—
251C22×C3—181no—
252C22×C3—181no—
253C22×C3—362no—
254C2×D3—21no—
255C2×D3—62no—
256C2×D3—61no—
257C2×D3—61no—
258C2×D3—61no—
259C2×D3—61no—
260C2×D3—121no—
261C2×D3—122no—
262C2×D3—122no—
263C2×D3#62611yescharacteristic
264C3×C4—21no—
265C3×C4—21no—
266C3×C4—21no—
267C3×C4—31no—
268C3×C4—31no—
269C3×C4—41no—
270C3×C4—61no—
271C3×C4C2×D3211yescharacteristic
272C3×C4C2×D3211yescharacteristic
273C3×C4C2×D3211yescharacteristic
274Dic3—182no—
275Dic3—181no—
276Dic3—181no—
277Dic3—181no—
278Dic3—361no—
279Dic3—361no—
280Dic3—362no—
281Dic3#62611yescharacteristic
282C32—21no—
283C32—21no—
284C32—21no—
285C32—41no—
286C32#90811yescharacteristic, complemented
287C32#91111yescharacteristic, complemented
288C32#93211yescharacteristic, complemented
289C23—181no—
290Q8—271no—
291D4—31no—
292D4—91no—
293D4—91no—
294D4—181no—
295D4—541no—
296C2×C4—31no—
297C2×C4—541no—
298C8—31no—
299C8—91no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—41no—
304C2×C3—61no—
305C2×C3—61no—
306C2×C3—121no—
307C2×C3—121no—
308C2×C3—121no—
309C2×C3—241no—
310C2×C3—361no—
311C2×C3—361no—
312C2×C3D3×D1211yescharacteristic
313C2×C3C2×#62611yescharacteristic
314C2×C3D3×#6711yescharacteristic
315D3—21no—
316D3—41no—
317D3—122no—
318D3—121no—
319D3—121no—
320D3—121no—
321D3—121no—
322D3—241no—
323C22—31no—
324C22—61no—
325C22—181no—
326C22—181no—
327C22—362no—
328C4—31no—
329C4—541no—
330C4D3311yescharacteristic
331C3—21no—
332C3—21no—
333C3—21no—
334C3—41no—
335C3#6379511yescharacteristic, complemented
336C3#6383011yescharacteristic, complemented
337C3#6393211yescharacteristic, complemented
338C2—61no—
339C2—121no—
340C2—361no—
341C2—361no—
342C2D3×#62611yescharacteristic
subgroupG/H
centerC2D3×#626
commutator, Fitting, cosocleC33×C4C23
Frattini, hypercenterC4D33
socleC2×C33C2×D4
hypocenterC33AGL(1,ℤ/8)
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C4

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #89021C2semidirect
2. C3×D3×D12C2semidirect
3. #88810C2semidirect
4. #88796C2semidirect
5. #88794C2semidirect
6. C3×#3714C2semidirect
7. C3×#3711C2semidirect
12. #6308C22semidirect
14. C3×C4⋊(C3⋊D3)C22semidirect
20. C3×C3⋊(C3⋊C8)C22semidirect
21. C32×C3⋊C8C22semidirect
25. D3×D12D3semidirect
43. #3714D3semidirect
45. #3711D3semidirect
64. C4⋊(C3⋊D3)C2×D3semidirect
95. C3⋊(C3⋊C8)C2×D3semidirect
100. C3×C3⋊C8C2×D3semidirect
101. C3×C3⋊C8C2×D3semidirect
169. C33AGL(1,ℤ/8)semidirect
214. C3⋊C8D32semidirect
286. C32#908semidirect
287. C32#911semidirect
288. C32#932semidirect
335. C3#63795semidirect
336. C3#63830semidirect
337. C3#63932semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#890212
C3×D3×D122
#888102
#887962
#887942
C3×#37142
C3×#37112
#639323
#638303
#637953